Vol.9, No.19, 137/ PID A Proposal of PID Auto-Tuning for a Class of Unsymmetrical Processes Kazuhiro Mimura and Tetsuo Shiotsuki In this paper

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1 Vol.9, No.19, 137/ PID A Proposal of PID Auto-Tuning for a Class of Unsymmetrical Processes Kazuhiro Mimura and Tetsuo Shiotsuki In this paper, we propose a new tuning method for PID parameter especially for a class of unsymmetrical processes. The advent of relay auto-tuning of PID controllers got rid of previous troublesome tuning procedure. However, when the process is unsymmetrical the ultimate gain and ultimate period obtained are no longer reliable since describe functional analysis that assumes sinusoidal input can t be applied properly. The proposed method uses pseudo-ultimate gain, pseudo-ultimate period, and heat time ratio obtained from conventional relay feedback method, which gives better PI parameter for a lamp heater circulator system than conventional tuning rule. The experimental test result showed proposed rule achieved less overshoot and quicker disturbance recovery. Key Words: PID control, relay auto-tuning, unsymmetrical process 1. Wang 3) Hägglund Tengvall 1984 Åström 1) PID Ziegler and Nichols PID PID Tan 4) PID Ziegler and Nichols Hägglund Wang Wang Hägglund Tengvall ) Tan 4 5 ICCAS-SICE ( )KELK Technical Department, KELK Ltd., Hiratsuka-shi, Kanagawa Dept. of Robotics and Mechatronics, Tokyo Denki University, Chiyoda-ku, Tokyo Received October 8, 009 TRIA 019/10/ c 009 SICE

2 ' 138 SICE T. on Industrial Application Vol.9 No Fig. PID PI. PID A r e u y Plant PID B Fig. Halogen lamp heater circulator Fig. 1 Block diagram of a standard relay auto-tuner Fig. 1 A Fig. 3 T c a Kc 1) ( % M K c, T c K c = 4M πa 1 Ziegler and Nichols +-,.(*) /0( % PID B PID Fig. 3 An example of relay auto-tuning of unsymmetrical PID response. Fig. 4 U max, U min ) U max U bal U bal U min Fig. 4(a) 1 (*) % & $ #

3 U bal 4. Fig. 4(b) U bal (a) symmetrical response 1) ) 3) 4) 5) 4. 1 Table1 (Fig. 5) V 1 ρc p dt b (t) dt <,= = P (t) qρc p (T b (t) T a (t)) 3 #%$ &(' ) *) : %,- ) * 6; 9:../10 10-)34 )34#5 Fig. 5 ) (Fig. 6) Halogen lamp heater dt a (t) V b ρc p = qρc p(t b (t) T a(t)) dt α 0V b ρc p(t b (t) T r(t)) 4 Fig. 4 (b) unsymmetrical response An example of relay auto-tuning of unsymmetrical response. P V :output temperature, u:manipulated variable -8 7 :9' &;& (<;& ;= ' & ( )*$$ &)*+,+, -.0/( ,$7 >?@AB?7C0D7> EFC0? G?,H G I C K cp T cp Fig. 3 )*<.+, )&K X = T h T cp -J+, )&K # #$% & 3 PI Fig. 6 Chemical bath

4 0 (% & % /). + ) +) * ' $, 140 SICE T. on Industrial Application Vol.9 No Table 1 Variables in the equation(3)(4) notation description unit T a temperature at the heater [ C] bottle inlet and bath outlet T b temperature at the heater [ C] bottle outlet and bath inlet P halogen heater power applied [W] to the chemical Table Parameters in the equation(3)(4) notation description unit V 1 heater bottle volume [m 3 ] V b chemical bath volume [m 3 ] ρ density of the chemical [kg/m 3 ] C p specific heat of the chemical [J/kgK] q flow rate of the chemical [m 3 /sec] α 0 heat loss coefficient [1/sec] *' ( 1+. / * + ) ' ( # &% #$ (a) Pseudo-ultimate gain Table α 0 P ss α 0 = 5 V b ρc p(t ss T r), - P ss T ss T r ( ) α 0 ' % & α 0 Fig. 7 (3),(4) P V T sen [sec] L[sec] (b) Pseudo-ultimate period (6) q α 0 V b ρc p G p (s) = (V 1 V b s + (V 1 + V b )qs + α 0 V b q) 1 T sens + 1 e Ls 6 Fig. 8 Comparison of pseudo-ultimate gain (a) and pseudoultimate period (b). X-axis shows the simulated result and Y-axis shows the actual experiment result Table Fig. 7 Whole plant block diagram Table 3 Some test conditions and levels condition level set point [ C] 15, 35, 55, 95, 145 chemical bath volume [L] 0, 40, 60 heat loss coefficient [1/sec] delay time [sec] 30, 60, ) 16 Fig. 8 MATLAB

5 Simulink ) PI J set J dis Kcp T X PI PI Kp Ki, J set = w 1OS + w IT AE m, n J dis = w 3IAE + w 4Ms 9. K p = K cp /m, m = 1.5,.0,.5,, 5.0 K i = T cp n, n = 0.50, 0.75, 1.00,, PI CHR Chien, Hrones, Rezwik PI Z and N(Ziegler and Nichols) PI Table 4 7) C(jω) M s = max 1 8 PI <ω<1 1 + P (jω)c(jω) M s < M s <.0 Fig. 9 1 PI X-Y Z ITAE ITAE x proportional gain level 6 Fig disturbance profile 4 Integral time level 4. 4 w1, w, w3, w4 PI Fig. 10 PI Table 4 Ziegler and Nichols, CHR tuning rule. Kc: ultimate gain, Tc: ultimate period tuning rule P I D Z and N: PI 0.45Kc 0.83Tc 0 ITAE IAE Z and N: PID 0.6Kc 0.5Tc 0.15Tc CHR: PI 0.3Kc 1.0Tc 0 M s 5) P (jω) CHR: PID 0.45Kc 0.6Tc 0.1Tc Fig. 11 X PI m, n PI X 1) K p = K cp/(a Kpe B Kp X ) K i = T cp (A Kie B Ki X ) ) K p = K cp /(C Kp X + D Kp ) K i = T cp (C Ki X + D Ki ) K p K i A Kp, B Kp, A Ki, B Ki, C Kp, D Kp, C Ki, D Ki Fig. 1 Fig. 10 PI PI

6 14 SICE T. on Industrial Application Vol.9 No Best PI CHR (a) Best PI Z and N Fig. 10 (b) Comparison of response at SV=55 [ C] (a) Set point response of best PI parameters (top) and CHR (bottom). (b) Disturbance response of best PI parameter (top) and Ziegler and Nichols (bottom). 55. Fig. 13 Fig , Fig. 14 Fig. 15 X = 0.0, 0.5, 0.50, 0.75 (b) Ziegler and Nichols CHR Fig. 1 Responses of the proposed method at SV=55 [ C]. Hägglund 6) AMIGOf(Approximate (a) set point response and (b) disturbance response MIGO based on frequency response data) AMIGOf (11) PI 135 X (10) PI m, n : scale factor m, n : scale factor Fig X : heat time ratio (a) X : heat time ratio (b) Relationship between the heat time ratio X and scale factor m, n. (a) set point response and (b) disturbance response (a) P I P I

7 Fig. 13 (a) PV: process variable (b) u: manipulated variable Set point responses of different response speed L/T = 1 9 CHR 3) T, L CHR Ziegler and Nichols L/T Fig. 16 PI 6. 1 (6) 3 PID (9) 1) 1 )L/T < 0.1 K T L 3) 1 K, T L/T K SV SV/K L/T SV/K 91 L/T = 0.001, 0.01, 0.0, 0.03, 0.04, 0.06, , 0.10, 0.11, 0.40, 0.75, 1.00 SV/K = 0.05, 0.05, 0.1, 0.5, 0.5, 0.75, 0.9 K T K = 00 T = 1600 Kp Ki u u = 1 u = 1 Kp Ki Kp Ki T L/T Fig. 17(a) (f) 1) L/T < 0.1 3% Ki/T > 1 T ) L/T > 0.1 SV/K

8 144 SICE T. on Industrial Application Vol.9 No %'&( %)(*+, -/.0 134,5 # $ %'&)( *,+ - $. / )8 # $&%'$(' ) *,+.- /10 ) ;:=<.> # $&%' (*) + #-,.0/ # (a) X = 0.0 $&%'($)'+*, -.0/ :<;>=@?ACB # #%$'& ( )+*-,.0/1 ( ;:<-= (c) X = 0.50 $%&(')*&+)-,./ # /9 :<;>=?A@CB # $ #% &(') *,+- %. / (b) X = 0.5 Fig. 14 (d) X = 0.75 Comparison of set point response between proposed tuning rule and other tuning rule with different heat time ratio. ZandN: Ziegler and Nichols rule, CHR: Chien, Hrones, and Reswik rule and AMIGOf: rule proposed by Hägglund and Åström.

9 %'&( %'( ) *,+.- /10 ) :9<;>= # $ % &('*) +-,. %/ :9 %'&( %'( ) *,+-.0/1 ) ;:=< # $ %'&( )+*, $-.0/ (c) X = 0.50 # (a) X = 0.0 $&%' $(' ) *,+.- /10 ) :9<;= # $ %'&( )+*, $ -.0/ %'&( %'( ) *,+-.0/1 ) :9<; # $&%' (*) + #, -/ (b) X = 0.5 Fig. 15 (d) X = 0.75 Comparison of disturbance response between proposed tuning rule and other tuning rule with different heat time ratio. ZandN: Ziegler and Nichols rule, CHR: Chien, Hrones, and Reswik rule and AMIGOf: rule proposed by Hägglund and Åström.

10 146 SICE T. on Industrial Application Vol.9 No %'&)(%*( )+,.-/10354,.-/ #%$& #'& ( )+*-,/ ( #%$&('*),+ #%$&('*),+.- %'&)(+*,.- / 10 &)(+* (c) SV=60[ C] # (a) SV=40[ C] $&%('$)' (* +-,.0/143 +-,.0/14365 #$ %$ &' (*)+-,.0/ 13 4 '65,.7/ # $ % &(')+*,.- &(')+*,.-$/ #%$&('*),+ -/.0 1 '*),+ (d) SV=80[ C] (b) SV=60[ C] Fig. 16 Comparison between proposed tuning rule and conventional tuning rule. (a)(b) set point response and (c)(d) disturbance response. ZandN: Ziegler and Nichols rule and CHR: Chien, Hrones, and Reswik rule

(a) L/T = 0.001 (c) L/T = 0.1 9 19 010 147 (b) L/T = 0.03 (d) L/T = 0.4 1983 199 001 KELK 1983 89 008 10 IEEE Fig. 17 (e) L/T = 0.75 (f) L/T = 1.

Hägglund: Automatic tuning of simple regulators with specifcation on phase and amplitude margins, Automatica, 0, 645/651 (1984) T. Hägglund and A.

11 (a) L/T = (c) L/T = (b) L/T = 0.03 (d) L/T = KELK IEEE Fig. 17 (e) L/T = 0.75 (f) L/T = 1.00 Comparison of proposed method (solid line) and best PI parameter (dot) 1 K. J. Åström and T. Hägglund: Automatic tuning of simple regulators with specifcation on phase and amplitude margins, Automatica, 0, 645/651 (1984) T. Hägglund and A. Tengvall: An automatic tuning procedure for unsymmerical processes, ECC 95, 450/455, Rome, Italy (1995) 3 Q. G. Wang, C. C. Hang and W. Zou: Automatic tuning of nonlinear PID controllers for unsymmetrical processes, Computers chem. Engng,, NO.4/5, 687/694 (1998) 4 K. K. Tan, Q. G. Wang, T. H. Lee and C. H. Gan: Automatic tuning of gain-scheduled control for asymmetrical processes, Control Engineering Practice, 6, 1353/1363 (1998) 5 K. J. Åström and T. Hägglund: PID Controllers: Theory, Design, and Tuning, nd ed., 15/17 ISA (1995) 6 T. Hägglund and K. J. Åström: Revisiting the Ziegler- Nichols tuning rules for PI control-part II The frequency response method, Asian journal of control, 6-4, 469/48 (004) 7 : PID,, pp167 (199)

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